I need to write a program that calculates the sum of all primes smaller than a given number $N$
($10^{10} \leq N \leq 10^{14} $).
Obviously, the program should run in a reasonable time, so $O(N)$ is ...

I'm regarding a stochastic process $(X_t)$of which the mean starts at $O(n)$ and is reduced by the factor $(1-r)$ in each step with $r = \Omega (1/n^9)$, so $$E(X_{t+1}) \leq E(X_t) (1-r) .$$
Now it ...

I am trying to learn about complexity theory, which states that $f(n)$ is in $O(g)$ if, for some $C > 0$, $f(n) \leq C\cdot g(n)$ for all $n \in \mathbb{N}$. That's well and good; it makes sense to ...

I'm having a bit of difficulty understanding big-omega and big-theta of this particular function which is supposedly Ω(16n + 33)
$5n − 2 = Ω(16n + 33)$
I understand that the there is some constant c ...

We know that all the coefficients $a_1, a_2, \ldots , a_m$ are integer. Also, $K$ is an integer number. I only need to know if the inequality has a integer solution or not. It means that there is no ...

We know that the chromatic number of a graph $G$ is the smallest number of colors needed to color the vertices of $G$ so that no two adjacent vertices share the same color .
But why the coloring is ...

This has puzzled me for a little. I start off with a list of primes that is sufficiently large. For my number $n$, I do trial division of primes in ascending order until I reach a prime that divides ...

Suppose I have an output of roughly $n^k k \log_{10} n$ digits where $A$ has $n$ elements and we have to list $n^k$ tuples with $k$ components each. What would be the time complexity class of such a ...

I'm trying to approximate a 1D definite integral to within an additive $\epsilon$ for a given $\epsilon$. I was wondering whether there is an $O(\text{polylog}(1/\epsilon))$-time algorithm for this. ...

So we know that :
(1). A problem is NP-complete if every other problem in NP can be reduced to it in polynomial time
(2). A problem is said to be strongly NP-complete if a strongly NP-complete problem ...

Is there some maximum degree for a polynomial for time complexity considerations and maybe P-NP considerations, maybe some high-degree polynomial formula identified by name, and associated with some ...

We have two arrays $A,B$ with sizes $n,m$ respectively. We know that $m \geq n$. We also know that no array contains the same number twice.
Propose an algorithm that prints how many numbers appear in ...

We have $2n$ values $x_1,x_2,x_3,\ldots,x_n$ and $y_1,y_2,y_3,\ldots,y_n$ such that the pair $(x_i,y_i)$ represents the location of a city $i$. Assume there is no straight line that goes through all ...

Currently I am learning ( a beginner ) about Bell inequalities and device independent outlook on quantum mechanics. I come across some papers using these concept in quantum game theory. Most of the ...

I am trying to derive the LU decomposition time complexity for an $n \times n$ matrix.
Eliminating the first column will require $n$ additions and $n$ multiplications for $n-1$ rows. Therefore, the ...

Here's the problem. Every word in a dictionary is defined by a set of other words. For example "cat" may be defined as "small mammal with fur". Can we choose a set of 'base' or 'prime' words such that ...

Appologies if this question is utterly naive. I know very little about complexity classes, but like to learn more.
Consider the following problem. Given input $n$ (a natural number) we want to find ...

I'm interested in the idea that highly selected causal systems exhibit general behaviors and properties. By causal systems I mean formal systems that progress from a defined set of starting conditions ...